Contract Negotiation and the Coase Conjecture:
A Strategic Foundation for Renegotiation-Proof Contracts ∗Bruno Strulovici
Northwestern University
December 8, 2016

Abstract
What does contract negotiation look like when some parties hold private information and negotiation frictions are negligible? This paper analyzes this question andprovides a foundation for renegotiation-proof contracts in this environment. The model
extends the framework of the Coase conjecture to situations in which the quantity or
quality of the good is endogenously determined and to more general environments inwhich preferences are nonseparable in the traded goods. As frictions become negligible, all equilibria converge to a unique outcome which is separating, efficient, andstraightforward to characterize.

∗

I am grateful for comments from Mehmet Ekmekci, Drew Fudenberg, Lucas Maestri, Jim Malcomson,

Paul Milgrom, Alessandro Pavan, Marcin Peski, Ariel Rubinstein, Larry Samuelson, and Asher Wolinsky,
discussions by Takuo Sugaya and Alex Wolitzky, as well as the editor and four anonymous referees, Mikhail
Safronov’s excellent research assistance, and reactions from seminar participants at Princeton University,
Caltech, UCSD, Oxford University, the École Polytechnique Fédérale de Lausanne, the Séminaire Roy inParis, the University of Illinois at Urbana-Champaign, the London School of Economics, the University ofEssex, the University of Pennsylvania, Duke University, the 2013 Cowles Conference in Economic Theory,
Kyoto University (KIER), the Stanford Institute for Theoretical Economics (SITE 2013), the Asian Meetingof the Econometric Society (2013), Columbia University, Western University, the University of Toronto, the2014 Winter Meetings of the Econometric Society, the Toulouse School of Economics, and NYU Stern. This
research is supported by an NSF Grant (No.1151410) and a fellowship from the Alfred P. Sloan Foundation.

1

1

Introduction

Real negotiations contain a puzzle: On the one hand, parties often try to withhold private
information until a deal is reached. On the other hand, the very fact of agreeing to a deal,
and the timing of this agreement, reveals to some extent the parties’ real stakes in thenegotiations. How is this new information incorporated in the final agreement, when players
can freely renegotiate their contract?
This puzzle does not arise in the best-known models of negotiation, because of their specific structure. In Coase’s model of a durable-good monopolist (Coase (1972)), for instance,
buyers need only one unit of the good. Any sale is therefore efficient, and further negotiations are pointless regardless of the information revealed about the buyer by the timing of
his purchase. Similarly, when players with privately known patience bargain over splitting apie (Rubinstein (1985)), any split of the pie is ex-post efficient and further negotiations arepointless regardless of what players learn about each other’s patience.
In richer contractual environments, however, an initial agreement may be inefficient. If,
for example, a buyer may value multiple units or various qualities of a good, his acceptanceto buy the good early in the negotiation process may reveal a preference for more units
or higher qualities of the good, spurring further negotiations before the final agreement.
Moreover, this perspective affects both parties’ incentives from the outset of negotiations.
This paper’s objective is to characterize the possible outcomes of negotiation in thesericher contractual environments, when the ability to refine contracts is (almost) unrestricted.
It provides a dynamic resolution of the above puzzle by describing the gradual succession ofagreements leading to the final outcome, and answers the following questions: How gradualare the agreements? How fast do they incorporate private information? How much efficiency
is lost in the process? How does the outcome depend on the buyer’s type and on the seller’sbelief about it? And on the initial relationship between these parties?
The analysis is based on an explicit protocol of negotiation in which one party—theagent—possesses private information while the other party—the principal—makes the offers.
The framework is broader than the buyer-seller model with quasilinear preferences and linear
cost. For example, the goods traded may be complements.
The main result is that all equilibria of this negotiation game converge to a unique out2

come, which is fully separating and efficient, as negotiation frictions become negligible. The
outcome is renegotiation-proof in the sense that no surplus may be gained from renegotiating
it further. Seen in this light, the paper provides a strategic foundation for renegotiation-proof
contracts, as well as a dynamic implementation, without commitment, of efficient allocations.
In the negotiation protocol considered here, the principal can propose new contracts orchanges to a previously accepted contract at any time, including after the agent has acceptedor rejected a previous proposal. The principal is thus unable to commit not to renegotiate thecontract. Time is divided into negotiation rounds: In each round, the principal can propose
various changes to the current agreement. The agent then accepts a proposal or rejects
all changes. This exchange captures in a stylized fashion the idea of a gradual agreement
formation in real negotiations; here, any agreement is interpreted as a (possibly oral) bilateralcontract, binding unless both parties agree to replace it by another contract.1
For the final outcome to be well defined, negotiations must end somehow, and the protocol
relies on a particular concept of negotiation friction: at the end of each round, negotiations
are exogenously interrupted with a fixed probability, η, in which case the current agreement
is implemented. This interruption may be interpreted in various ways. For example, supposethat parties are negotiating a risk-sharing contract, each dimension of which concerns a stateof the world. Then, the realization of the state of the world (or its public announcement)
makes any further negotiation moot. In a sales contract application, interruption may be
coming from a third party, supplier or customer, demanding a commitment or service which
requires the immediate implementation of the contract. The interruption probability captures the negotiation friction. When η is equal to 1, the protocol reduces to full commitment
since the first proposal is also the last one. In this case, it is well-known that the principal
typically distorts the allocation of some types of the agent, causing ex post inefficiency. This
paper’s interest lies in the opposite case, in which negotiation frictions are negligible (η goesto 0). This case should be interpreted as parties having arbitrarily frequent opportunities
to negotiate with each other: the time interval between consecutive rounds is so small thatparties become unlikely to be exogenously interrupted in any such interval. Under this inter1

In practice, reneging on an agreement is costly even if it was made orally or informally—indeed, many

jurisdictions recognize oral contracts as legally binding. Reneging on an informal agreement bears othercosts, as it damages the reputation of the reneging party and puts a strain on further negotiations.

3

pretation, the time interval between rounds is of order η. Therefore, even though it takes anincreasing number rounds for negotiations to stop as η goes to zero, the expected stopping
time of negotiation is independent of η and thus uniformly bounded.2
While the ability to freely modify past agreements seems necessary to guarantee an ex
post efficient outcome, establishing that it is sufficient involves complex issues. To appreciate the difficulty, consider again the standard durable-good monopolist. Efficiency in that
context means that the good is sold without any delay, and was established by Gul, Sonnenschein, and Wilson (1986) as the discount rate between consecutive periods goes to zero.3
The proof is sophisticated even in this considerably simpler contractual environment, where
each contract amounts to a single posted price. The key question is to determine whether
the seller can benefit from distorting the allocation of the low-valuation buyer by inefficiently
delaying the sale, in order to extract some rent from the high-valuation buyer. In richer environments, the question is more complex because i) any initial agreement may be followed byfurther negotiations (e.g., contractual covenants, increases in quantities or qualities), ii) the
principal may benefit from proposing multiple new contracts at each round instead of asingle one,4 iii) the agent can randomize over all such contracts resulting in complex belief
dynamics, and iv) utility may be nonlinear and non-separable in the contract components.
Due to the complexity of the analysis, the model focuses on a binary information struc2

Precisely, suppose that the time interval between two rounds is equal to a × η for some a > 0. Thenthe interruption time has mean a regardless of η and becomes approximately exponentially distributed withparameter 1/a as η goes to zero.
3
The result is shown for the “gap” case and the “no gap” case under some Lipschitz condition on the
distribution of types, for weak Markov equilibria (see also Sobel and Takahashi (1983) and Fudenberg, Levine,

and Tirole (1985)). Ausubel and Deneckere (1989) show that the conjecture can fail when more generalequilibria are allowed. The analysis of the Coase conjecture has been extended to various environments:
interdependent values (Deneckere and Liang (2006)), an incoming flow of new buyers (Fuchs and Skrzypacz(2010)), and outside options for the buyer (Board and Pycia (2013)). In Board and Pycia, the seller can
extract all the surplus owing to positive selection, unlike the present model, which is closer to negative
selection. Skreta (2006) takes a mechanism design approach and shows the optimality of price posting.
All these papers assume that the buyer can buy only one unit of the good, a single quality of the good isavailable, and utility functions are quasilinear.
4
For example, the principal may propose one contract for each type of the agent, or propose multiple
almost identical contracts as a communication device to emulate cheap talk.

4ture: the agent can be of two types, and the corresponding utility functions satisfy a standard
single-crossing condition. As long as the types of the agent have not been fully separated,
there are strictly positive gains from renegotiation, and the paper’s main result shows that
all these gains are realized as frictions become negligible: all Perfect Bayesian Equilibria(PBEs) converge to a unique outcome, which is separating and efficient.
The type-specific contracts to which all PBE outcomes converge are straightforward tocharacterize and determine graphically. Unlike the full-commitment case (but like the Coase
conjecture, which it generalizes in this respect), these contracts are independent of the initial(non degenerate) belief that the principal holds about the type of the agent. They do dependon the initial contract—or absence thereof, which is formally equivalent to a ‘null’ contract—which may lie in three possible regions of the contract space. In the “No-Rent” region, theprincipal extracts all surplus of renegotiation regardless of the agent’s type. In the othertwo regions, there is one region-specific type (“L”, say) who gains nothing from negotiation
while the other type (“H”) gets a positive rent.
The fact that almost-efficient contracts are proposed immediately implies that renegotiation plays a relatively minor role in equilibrium, even though the possibility of renegotiation
has a major impact on the outcome. This suggests that, empirically, one should not inferthat renegotiation is impossible or difficult in practice just because the observed renegotiation activity seems negligible. Instead, negotiation may be feasible and cheap but largely
internalized in the very first contracts that are proposed.
Section 2 presents the setting and main results. Section 3 compares the results with the
Coase conjecture. The main arguments are given in Sections 4 and 5. Section 6 discusses
the related literature and extensions of the framework.

2

Setting and Overview of the Results

There are two players, a principal (P) and an agent (A), who negotiate a contract lying insome compact and convex subset C of R2 whose components are denoted x1 and x2 .

A has a utility function uθ : C → R, where θ ∈ {L, H} denotes his type, and P has a costfunction Q : C → R. The functions uL , uH , and Q are twice continuously differentiable andhave strictly positive derivatives with respect to x1 and x2 ; uL and uH are concave and Q is5

x2
Px̄2 = xA
2 + x2

EH
EL

CH

CL

H

A
R0 = (xA
1 , x2 )x1
Px̄1 = xA
1 + x1Figure 1: Setting (trade interpretation)
convex. Although it is convenient to think of the agent’s type as being “high” (H) or “low”(L) to connect the results to the existing literature, the role played by each type is in fact
determined by the initial contract, as explained below.
A contract C = (x1 , x2 ) ∈ C is θ-efficient if it minimizes the principal’s cost among allcontracts in C providing θ with some given utility level. For each θ, let Eθ denote the setof θ-efficient contracts in the interior of C; θ’s iso-utility curve and P’s iso-cost curve aretangent at any such contract. It is assumed that the efficiency curves EL , EH are smooth andupward sloping.5 The setting is represented on Figure 1 in the context of a trade application(other applications are described later in this section). It is also assumed that, given any
contract C on Eθ , P’s and θ’s indifference curves going through C do not both have a zerocurvature at C. The assumption is satisfied, for instance, when the agent has a quasi-linear
utility function and his valuation for the good is strictly concave.6
5

These assumptions hold, e.g., if uH , uL , and −Q are weakly supermodular in (x1 , x2 ) and either uθ ’s or−Q is strictly concave in x2 , as explained in Appendix J. Substantively, these assumptions holds as long as
the goods are not too strongly substitutes of each other.
6
Alternatively, the agent’s utility could be linear in the good if P’s cost function is strictly convex. The
assumption does rule out settings in which both A’s and P ’s indifference curves are linear, but even thiscase can be approximated by an arbitrarily small curvature (cf. Section 3). The assumption guarantees that

6

The functions uL and uH are required to satisfy a standard single-crossing condition:
iso-utility curves of L are steeper than those of H at their intersection point. This implies
that the efficiency curve EL lies to the lower right of EH . C can therefore be partitioned into
three regions separated by EL and EH . Contracts in the inner region are said to be in the ‘No
Rent’ configuration, while contracts strictly below EL (above EH ) are in the ‘H-Rent’ (‘L-Rent’) configuration. The set of contracts in the H-Rent configuration is denoted by H. In
the trade application, C represents an Edgeworth box, delimited by the sum of endowmentsof the agent and the principal. A contract C specifies the final allocation of the agent, theefficiency curve Eθ is the ‘contract curve’ corresponding to type θ, and the status quo R0
represents the endowment of the agent before any trade.

The Negotiation GameThe game unfolds as follows. First, the agent privately observes his type θ; P has aprior characterized by the probability β0 = P r(θ = H). The game starts with a reference“contract” R0 ∈ C, which represents the current engagement between the principal and theagent. In many settings this initial contract would simply represent the absence of any pastengagement, as in most contracting models. In the durable-good monopolist application,
the initial contract is the “no sale” outcome; in the trade application, the initial contract
corresponds to the initial endowment of the agent before any trade. Specifying R0 explicitly
is useful for two reasons: First, it will allow us to treat the initial round like any later roundin which an agreement has already been made, which simplifies the exposition. Second,
there are environments, such as the trade application, in which the initial contract (initialendowment of the agent) plays an important role on the outcome of negotiations.
There are countably many potential rounds, indexed by n ∈ N. At each round n, P canpropose a finite menu Mn of contracts in C. In terms of interpretation, proposing contracts orchanges to the current contract is formally equivalent; the former formulation is used here.7

the distance between the two curves increases quadratically as one moves away from C, a property used tocompute a lower bound for the inefficiency of some contracts (see Lemma 15).
7
One could restrict the principal to propose at most two contracts in each round. Such a restriction is notdesirable for several reasons. First, there is no guarantee that proposing only two contracts at each roundis without loss of generality. As Bester and Strausz (2001) have shown, the set of implementable outcomes

7

The agent chooses a contract in Mn or holds on to the last accepted contract, Rn . Any mixed
strategy over the choice set Mn ∪ {Rn } is allowed. The selected contract, Rn+1 , becomes the

new reference. At the end of each round, negotiations are frozen with probability η ∈ (0, 1],
in which case the last accepted contract, Rn+1 , is implemented. Otherwise, negotiations
move on to the next round. The event of a negotiation freeze will be hereafter referred toas a “breakdown.” This term means that future negotiations are terminated. However, thelast agreement formed before the breakdown (or the initial contract, if no agreement wasformed) is still valid. As explained in the Introduction, one may view η as the time interval
between two rounds, in which case the expected stopping time of negotiation is independentof η and thus bounded as η goes to zero.
Letting {Rn } denote the stochastic process of contracts entering each round n, the agent’s"#
expected utility is equal toXVθ = E(1 − η)n ηuθ (Rn+1 )n≥0while P’s expected cost is
Q=E

"
X
n≥0

#

(1 − η)n ηQ(Rn+1 ) .

The parameter η represents the negotiation friction of the game.8 The objective of thispaper is to characterize the PBEs of the game as η goes to zero. The existence of a PBE isguaranteed by Theorem 1, whose proof is in the Online Appendix (Appendix D).
Theorem 1 For each η ∈ (0, 1], there exists a PBE of the negotiation game.
To prove this theorem, backward induction techniques cannot be used because there is nonon-degenerate belief, no matter how extreme, for which negotiations end in finite time, as
generally requires strictly more “messages” (or contracts) than the number of types of the agent, even in a
two-period setting. Here, one must consider all possible continuation equilibria, including incentive inefficientones. Indeed, an inefficient continuation equilibrium may provide incentives at earlier stages of the gameand it is precisely the purpose of the analysis to show that the negotiation outcome is efficient, rather thanassume this. Second, such a restriction would not necessarily simplify the analysis as the agent would stillchoose between three contracts (the two contracts offered and the current one), which potentially results inall the issues, such as belief non-monotonicity, which arise when more contracts are allowed. Finally, allowing
more contracts paves the way for an extension to three or more agent types, as explained in Appendix L.
8
There is another interpretation of the setting where η is the discount rate and the parties receive payoffs
at each period of the on going relationship. This interpretation is discussed in Section 6.

8

x2
x̄2

EH
No-Rent

L-RentEL

EH (R)R

EH (R̂)EL (R)EL (R̂)R̂

H-Rentx̄1(0, 0)

x1

Figure 2: Renegotiation outcomes
will be explained in the analysis below and proved in Appendix K. Instead, the proof builds
on a theorem by Harris (1985) for games of perfect information to show the existence ofan equilibrium in an auxiliary game between the principal and the high type of the agent,
which treats the low type “mechanically.” This equilibrium is then used to construct anequilibrium of the negotiation game with private information.
For any contract R ∈ C, let EH (R) and EL (R) denote the least costly pair of H- andL-efficient contracts such that each type θ 6= θ′ weakly prefers Eθ (R) to Eθ′ (R) and to R.

This pair is well defined for each possible configuration of R.9 Figure 2 represents thesedefinitions for the case of CRRA utility functions and a linear cost function.
Theorem 2, below, requires that no contract arising in equilibrium be jointly efficient9

If R is in the No-Rent configuration, Eθ (R) is simply the θ-efficient contract that gives θ the same utility

as R. If R is in the H-Rent configuration, then EL (R) is similarly defined, while EH (R) is the H-efficient
contract that gives H the same utility as EL (R). Because that contract gives a strictly higher utility to H
than the initial contract R, H must be getting a positive rent in any equilibrium, hence the name of that
configuration. A symmetric construction obtains if R is instead in the L-Rent configuration.

9

for both agent types. The single-crossing property already rules out such contracts in theinterior of C. To deal with C’s boundary, say that a contract R0 is regular if it is in the NoRent configuration or if it satisfies the following condition, stated for R0 ∈ H (an analogous
condition is required for the L-Rent configuration): for any R′ ∈ H,
uH (EL (R′ )) ≥ uH (R0 ) ⇒ EL (R′ ) 6= EH (R′ )(1)Given any contract R0 , regularity—whose role is explained below—can always be achievedby arbitrarily small perturbations of the utility or the cost functions, as illustrated by Section 3.10 In Figure 2, all contracts are regular except for the origin.
Theorem 2 Consider any regular contract R0 , belief β0 ∈ (0, 1), and ε > 0. There existsη̄ > 0 such that the following statements hold for any η ≤ η̄ and corresponding PBE:
A: The expected utility of each type θ is bounded below by uθ (Eθ (R0 )) − ε.

B: The probability that each type θ gets a contract within a distance11 ε of Eθ (R0 ) whenrenegotiation breaks down is greater than 1 − ε.
Theorem 2 shows that all PBEs reduce to an essentially unique one, which is efficient, as η

becomes arbitrarily small. The proof identifies a feasible strategy for P, which is to “giveup” on screening H, and proceeds to show that there is essentially nothing more that P cando. Methodologically, it is interesting to contrast this result with Ausubel and Deneckere(1989) who show that a large range of utilities and profits can be sustained in a Coase
conjecture environment with zero gap by punishing any seller deviation with a zero-screening
equilibrium. What serves in their setting as a punishment threat for the seller becomes herea tempting deviation for the principal, which prevents the existence of (essentially) any otherequilibrium. This model is thus closer to the positive-gap case of the Coase conjecture but
the gap here is endogenous and vanishes asymptotically, creating several major challenges
for the analysis explained in Section 5.
Theorem 2 implies that P always extracts some surplus from negotiation: When R0 is in
the No-Rent configuration, P extracts, in fact, all the surplus regardless of the agent’s type.
10

These arbitrarily small perturbations are chosen so as to slightly push efficient contracts on C’s boundary

into the interior of C, by an arbitrarily small amount.
11
The statement holds for any norm on R2 .

10

When R0 is in the H-Rent configuration, P extracts all the surplus from negotiating with L,and extracts some additional surplus if he faces H, corresponding to a move from EL (R0 ) toEH (R0 ).
Regularity is used as follows: When R0 ∈ H, one may show that any contract R′ ∈ Harising in equilibrium satisfies uH (EL (R′ )) ≥ uH (R0 ) (Proposition 1, Part iv) and, hence,that the premise of (1) is always satisfied on path. If R0 is regular, this implies that any

breakdown with a non-degenerate belief yields some inefficiency.
Applications
1. Durable-Good Monopolist. A is a buyer with quasi-linear utility uθ (C) = θū(x2 )+x1 , where x2 is the quantity of the good sold by P, x1 is A’s wealth, and ū is a strictly concave
function.12 The initial contract, R0 , is equal to (x̄1 , 0) where x̄1 is A’s initial wealth. P’s
cost is Q(x1 , x2 ) = cx2 + x1 , where c > 0 is the marginal cost for producing the good and x1
captures how much wealth P “leaves” to A.13
2. Labor Contract. P is a potential employer and A is a worker. −x2 represents A’seffort and x1 , his wage. A gets a utility uθ (C) = θψ(−x2 ) + x1 from contract C, where ψ

is a factor entering A’s cost of effort, increasing in its argument, and θ is a worker-specificskill entering his cost. The status quo R0 = (0, 0) represents unemployment. P’s profit isΠ(x1 , x2 ) = −Q(x1 , x2 ) = −x2 p − x1 , where p > 0 is the unit price of the good.

3. Consumption Smoothing and Insurance. There are two periods and a single

good. The dimensions of C represent A’s consumption in each period. P is a social planner

or a bank who can help the agent smooth his consumption. The agent’s type corresponds toa privately known patience/discount factor, or a distribution parameter describing how likely
the agent is to value the good in the second period. For example, u(x1 , x2 ) may be equal
to v(x1 ) + θv(x2 ) or to v(x1 ) + E[w(x2 , ρ̃)|θ] where ρ̃ is a taste shock whose distribution isincreasing in θ in the sense of first-order stochastic dominance and where w is supermodular,
so that E[∂w/∂x(x2 , ρ̃)|θ] is increasing in θ.14 R0 is A’s autarkic income stream. Q(x1 , x2 ) =12

The iso-level curves of uθ have a positive curvature as long as the second derivative of ū is strictly

negative, as is easily checked. A similar observation applies to ψ in the labor contract application.
13
P’s profit is Π(t, x2 ) = t − cx2 , where t is how much the agent pays P. Letting t = x̄1 − x1 , yields theformulation in terms of the cost function Q.
14
This application is explored in detailed by Strulovici (2013).

11

p1 x1 + p2 x2 , where pt is the market price for the good in period t.
4. Risk Sharing Each dimension corresponds to a state of the world. The quantity xi
specifies the transfer of a good from P to A if state i is realized. L values the good more instate 1 than state 2, relative to H. Alternatively, the types have the same preferences but
have different subjective beliefs, with L assigning a higher probability to the first state of
the world than H does.
5. Trade. More generally, the model describes a trade environment in which the dimensions of C represent distinct goods, with xi denoting the quantity of good i consumed by

A. Type L cares more about the first good than the second, relative to H. P (like A) has

convex preferences and Q is the negative of a utility function representing P’s preferences.
R0 denotes A’s initial holdings of the goods.

3

Relation to the Coase conjecture

In the standard Coase conjecture, all buyer types have a single-unit valuation. The set of
efficient contracts is the same for all types: it consists of all contracts for which the buyer
gets the good, regardless of the sale price.15 However, when goods are divisible or available
in multiple qualities, it becomes rather restrictive to assume that the same contracts which
are efficient for one type of the agent are also efficient for the other types. In fact, whenefficient contracts lie in the interior of the contract space, the strict single-crossing property
implies that whatever contract is efficient for one agent type is inefficient for the other.
This distinction explains why, in this paper, the principal can always extract some surplus
from the “high” type, in contrast to the Coase conjecture. It also explains why, when theinitial contact lies between the efficiency curves of both types—an impossibility for the
standard Coase conjecture, where these curves coincide—the principal can extract all thesurplus from negotiation. In this sense, the Coase conjecture appears to be non-generic as
it relies on the assumption that all agent types share the same efficient outcomes.
However, because the model imposes no lower bound on how “far” the efficiency curvesof both types have to be, it is easy to recover the Coase conjecture as a limit of Theorem 2.
15This statement concerns the “positive gap” case (lowest buyer valuation exceeds seller cost), which is

the relevant comparison here.

12

To see this, suppose that the first contractual dimension represents the agent’s wealth andthe second dimension represents the quantity, between 0 and 1, of a divisible good sold tothe agent. The initial contract is (W, 0), where W is the agent’s initial wealth. An agent oftype θ has utility u(x1 , x2 ) = vθ x2 + x1 with vH > vL > 0. The principal incurs a marginal∂Q(x1 , x2 ) = c(1 − x2 )δ−1 for producing a quantity x2 of the good, where c < vL andcost ∂x2δ ∈ [0, 1]. When δ = 1, the marginal cost is constant equal to c. The parameter δ is chosen
arbitrarily close to 1, so that Q is slightly convex with a strictly positive curvature.

The efficiency curve EH is horizontal and characterized by the quantity xH
2 (δ) solving

δ−1vH = c(1 − xH
. Likewise, EL is characterized by the quantity xL2 (δ) such that vL =2 (δ))c(1 − xL2 (δ))δ−1 . We havexL2 (δ) = 1 −



c
vL

1
 1−δ

and

xH
2 (δ) = 1 −



c
vH

1
 1−δ

,

so x2 (δ, L) < x2 (δ, H) and both converge to 1 as δ goes to 1. The efficiency curves of bothtypes are thus distinct but converge to the same contract curve, characterized by a single
unit of good sold to the agent, as δ goes to 1. The setting is represented on Figure 3. Red(blue) curves represent the iso-utility curves of the high (low) type. The boundary of regularcontracts is shown on the left of the figure. All contracts of H lying to the right of thisboundary, including R0 , are regular and if necessary one could expand the contract space tothe left (or translate the agent’s wealth) to make any given contract R ∈ H regular.

The Coase conjecture is recovered as follows: if P were sure to face H, he would move tothe contract CH on Figure 3. With uncertainty about the buyer’s type, however, Theorem
2 implies that the outcome is given by the contracts EH (R0 ), EL (R0 ), which converge to thesame contract as δ goes to 1. Both types of the buyer obtain essentially the same outcome,which is the (almost) sure sale of the good at the same price. The high type gets a rent
corresponding to the distance between EH (R0 ) and CH , while L gets no rent.
Theorem 2 shows which part of the Coase conjecture is robust to a richer contractual
environment: the efficiency part continues to hold, but the seller’s inability to extract any
surplus disappears. The richer contract space also gets rid of the stark discontinuity arisingin the Coase conjecture between the gap and no gap cases: With two types, H’s rent increases
as L’s valuation vL (and hence the equilibrium price) becomes lower. However, when L’svaluation reaches P’s marginal cost c, turning into the “no gap” case, H’s rent suddenly
13

1
CH

xH
2 (δ)

EH (R0)EL (R0)xL2 (δ)

FH
FL

B ou
nda
ry o
f re g
u lar
tra c
co n

H

ts
final wealth
R0 = (W, 0)

Figure 3: Recovering the standard Coase conjecture
drops to zero and P’s profit leaps up from zero to β0 (vH − vL ). Consider a similar exercise

in the setting of Theorem 2, where L’s efficiency curve is lowered until it goes through R0 .
The surplus that P extracts from H then varies continuously, until EL goes exactly throughR0 and P extracts all surplus from H.

4

Results holding for all friction levels

This section presents results holding for all values of η which will be used to prove Theorem 2.
When P assigns probability 1 to either type of the agent, or when the contract is in the NoRent configuration, there is a unique continuation PBE: P immediately extracts all the rent
from negotiation and efficiency obtains exactly.16 In other cases, one may compute an upperbound on the rent which the mimicking type (i.e., H, if R0 ∈ H) can extract. Intuitively, therent cannot exceed what this type would get if P “gave up” on screening H by immediately

giving the other type (L) his efficient contract. These results are collected in Proposition 1.
16

Showing that this outcome constitutes an equilibrium is straightforward, but establishing uniqueness ismore involved. The proof uses a cost undercutting argument similar to the one applied to utility levels in
Rubinstein’s (1982) bargaining model with complete information.

14

Unless indicated otherwise, all results of this section are proved in Appendix B.
Proposition 1 The following holds for any η and corresponding PBE:
i) If the prior β puts probability 1 on some type θ, P immediately proposes the θ-efficient
contract that leaves θ’s utility unchanged, and θ accepts it.
ii) If R0 is θ-efficient, P immediately proposes Eθ′ (R0 ) (θ′ 6= θ), and θ′ accepts it.
iii) If R0 is in the No-Rent configuration, P immediately proposes EL (R0 ) and EH (R0 ), andeach type θ accepts Eθ (R0 ).
iv) If R0 is in the H-Rent (L-Rent) configuration, H’s (L’s) expected utility is bounded above
by uH (EH (R0 )) (uL (EL (R0 ))).
From now on, the analysis focuses on R0 in the H-Rent configuration, the L-rent configuration case being symmetrical. The next result plays a crucial role for the analysis: At
any round n, P can propose the contracts EH (Rn ) and EL (Rn ) and have them accepted bytypes H and L, respectively. This deviation puts an upper bound on P’s continuation cost
as a function of the current contract and belief, and will often be referred to as “jumping”
or “giving up” (on screening H). Let βn denote the probability, at the beginning of roundn, that P assigns to type H.
Lemma 1 (Jump) If Rn is in the H-Rent configuration and P proposes the contracts EH (Rn )and EL (Rn ), with EH (Rn ) augmented by an arbitrarily small amount ε > 0, then H accepts
EH (Rn ) with probability 1 and L accepts EL (Rn ) with probability 1. Therefore, P’s continuation cost is bounded above by βn Q(EH (Rn )) + (1 − βn )Q(EL (Rn )).
The next result shows that as long as H hasn’t revealed himself, on-path contracts Rn alllie in H.
Lemma 2 For any R0 ∈ H and PBE, L accepts only contracts in H throughout negotiations.
Thus, any accepted contract Cn ∈/ H reveals H and can be replaced by the H-efficient

contract C̃n that gives H the same utility: this reduces P’s cost without affecting anyone’s

incentive. The remainder of the analysis focuses without loss on PBEs in which P only
proposes, in each round, contracts in H and/or the H-efficient contract giving H his con-tinuation utility. Given any PBE, any contract sequence {Rn } that is accepted by L with15

positive probability will be called a choice sequence. In equilibrium, the agent follows achoice sequence until, possibly, accepting an H-efficient contract, revealing by that choicethat he is of type H. Choice sequences have several important properties, described next.
Let uH (n) denotes H’s continuation utility at the beginning of round n and wn =uH (EH (Rn )) − uH (n). wn is the equilibrium rent that P takes away from H comparedto immediately giving up on screening him: it is a rent reduction index. But, as we shall see,wn is also closely related to the L-inefficiency of the current contract: the more L-efficient
Rn , the smaller wn has to be and wn = 0 if Rn lies exactly on EL .
Proposition 2 Along any choice sequence {Rn }, i) βn converges to zero, ii) Rn converges
to an L-efficient contract, and iii) wn converges to zero, as n goes to infinity.

Proposition 2 shows that full efficiency and screening obtain asymptotically. However, because negotiations break down exogenously in finite time, the key is to determine the speed
at which negotiated contracts converge to efficiency relative to the speed at which the exogenous breakdown occurs. In the standard Coase conjecture with two types, one may computea uniform upper bound on the time at which the sale takes place, which implies equilibrium
efficiency as the discount rate goes to zero. However, efficiency cannot be reached in finite
time here, as the next result shows (the proof is in Appendix K).
Proposition 3 For any R0 ∈ H, η ∈ (0, 1], n ∈ N, and PBE starting with belief β > 0, Rn
is not L-efficient. Equivalently, wn > 0 for all n.

Intuitively, this result comes from the richness of the contract space, and more particularly
the fact that negotiated contracts can become arbitrarily close to being L-efficient. In the
Coase conjecture with two types, there is a belief threshold β̂ below which the benefit ofscreening H becomes negligible relative to the cost of delaying a sale to L, prompting theseller to immediately set the price at L’s valuation. Here, instead, P can always propose acontract at a small distance x from the L-efficiency curve, which entails an L-inefficiency oforder x2 but provides a screening benefit of order x for H. For x small enough, this departurefrom efficiency is always beneficial.
If type H is very unlikely, however, P’s expected cost cannot be too different from what
he would get if he just ignored H. As a result, P leaves most of the rent to H, which puts
the following bound on wn .
16

1/3

Lemma 3 There exists Kw > 0 such that wn ≤ Kw βn

for all n, η, and PBE.

This bound can be used to prove Theorem 2 for some parameters of the model, but it is toocoarse for a general proof. Intuitively, Lemma 3 exploits only the inefficiency loss that P
incurs conditional on facing L while trying to screen H—a loss which is of second-order if R
is almost L-efficient. The general proof also exploits the loss incurred conditional on facing
H, which is of first order, and shows how these two kinds of losses interact and cause P togive up on screening H.

5

Proof of Theorem 2This section proves that the expected rent, w0 , which P takes away from H relative to H’sutility upper bound uH (EH (R0 )), converges to zero as η goes to zero. The other claims ofTheorem 2 are relatively simple corollaries of this result and are proved in Appendix I.

Part I: Block Construction
The proof starts by constructing blocks of rounds such that i) within each block, H isscreened with significant probability and ii) wn shrinks geometrically across blocks. The
construction draws from the dynamic screening literature17 but presents specific challenges.
The first challenge is that the inefficiency caused by a breakdown is endogenous (unlike,e.g., delaying a sale in the Coase conjecture when the gap is positive). Moreover, contracts
become asymptotically L-efficient conditional on staying in H, which makes the loss con-ditional on facing L arbitrarily small and thus hard to exploit. These contracts do entail
some non-negligible H-inefficiency, since they are bounded away from EH , but a contract

Rn ’s H-inefficiency only translates into a loss for P, relative to jumping to EH (Rn ), if theutility that Rn gives H is sufficiently close to what H gets from EH (Rn ). To ensure this, theproof uses the following observation: as a contract’s L-inefficiency becomes arbitrarily small,so does the maximum gain from screening H (both quantities are closely related to wn ).
Therefore, H-inefficiency must dominate the gain from screening provided that wn is small
17

In addition to Gul et al. (1986), see, e.g., Myerson (1991), Abreu and Gul (2000), Abreu and Pearce(2007), and Atakan and Ekmekci (2012, 2014).

17enough. The next result captures this intuition. Results of Part I are proved in Appendix F.

Lemma 4 There exist ε and D positive such that the following holds: if wm ≤ ε, then
Q(Rn ) ≥ Q(EH (Rm )) + D for n ≥ m.

Thus, even if Rm gives H a lower utility than the efficient contracts EH (Rn ) arising fromround m onwards, it is uniformly more costly to implement Rm than these contracts. Of
course, the lemma applies to round 0 only if w0 ≤ ε. However, if Theorem 2’s conclusion

holds for this case, the case w0 > ε can be easily ruled out by a discontinuity argument.18

Another challenge is that the inefficiency loss identified above is only incurred whenfacing H, but the probability of facing H becomes increasing smaller over time, conditionalon staying in H. This implies, as explained below, that a standard block construction witha fixed number of rounds in each block will not guarantee a geometric decrease in posteriors
across blocks. To induce this geometric decrease, we need to offset the rather ineffective
loss identified above by a sharper upper bound on P’s gain from screening H. This isachieved by controlling H’s continuation utility at the beginning of each block: the higher
H’s continuation utility at the next block, the lower P’s expected gains from screening H
from that block onwards.
Thus suppose that w0 ≤ ε, and start Block 1 at round n0 = 0. The size of Block 1 isdetermined endogenously: Let û0 = uH (n0 ), ê0 = uH (EH (Rn0 )), and β̂0 = βn0 . Also defineû1 by ê0 − û0 = t(û1 − û0 ), for a parameter t > 1 to be set shortly, and let n1 = inf{n :uH (n) ≥ û1 } denote the first round19 at which H’s continuation utility exceeds the thresholdû1 . The first round of Block 2—and, hence, the size of Block 1—is set to n1 .

The number of rounds in Block 1 must sufficiently large to guarantee that the probabilityof a breakdown and, hence, P’s expected loss during the block, is significant. This number
can be bounded below using H’s continuation utility, which increases by steps of order ηbetween consecutive rounds. Letting uH (n) denote H’s continuation utility at the beginning
18

Intuitively, wn ’s decrements are of order η. If one shows (for any m) that wm ≤ ε ⇒ wm = O(η), it

follows that w0 cannot exceed ε because the sequence wn would otherwise have to drop by an impossibly
large amount when it crosses ε. See Lemma 20 for the proof.
19
Lemma 18 guarantees that H’s continuation utility always reaches û1 in finite time.

18

of round n and ∆H = maxC,C ′ ∈C uH (C)−uH (C ′ ), H’s Bellman equation implies the following
result.
Lemma 5 uH (n) is nondecreasing in n and satisfies uH (n + 1) − uH (n) ≤ η∆H .
Therefore, reaching û1 requires at least n(1) = ⌊(û1 − û0)/η∆H )⌋ rounds.
¯
To guarantee that significant screening takes place during any given block and, hence,that P’s posterior goes down by a suitable factor, another challenge is to bound P’s gain
conditional on successfully screening the agent. This gain is closely related to the rent wnthat P takes away from H: Lemma 13 shows the existence of a Lipschitz constant a bounding
this gain by awn .20
These observations deliver a bound on the probability µ̂0 that H rejects all H-efficient
contracts until round n1 . Indeed, at round n0 , P can always implement the contracts(EH (Rn0 ), EL (Rn0 )), by Lemma 1. For this to be suboptimal, the gain from reducing H’srent through screening must outweigh the inefficiency loss from a negotiation breakdown,
which yields the following condition.
Lemma 6 µ̂0 satisfiesβ̂0 (1 − µ̂0 )a(ê0 − û0 ) + β̂0 µ̂0 a(ê0 − û1) ≥ β̂0 µ̂0 D

û1 − û0
.
∆H

(2)

The last term on the left is of particular importance: it is an upper bound on P’s expected
gain, relative to jumping at round n0 , from reducing H’s rent after round n1 . β̂0 µ̂0 is theprobability of facing H and H staying in H until round n1 , and û1 is the smallest utility

that P must provide to H at any round following n1 . This term is where the particularconstruction pays off: it captures the fact that P’s maximal gain from screening H decreases
across blocks.21
20

Lipschitz constants are used in the bargaining and reputation literature (see, e.g., Cripps, Dekel, andPesendorfer (2005)) where they arise naturally from the polyhedral structure of the utility sets. By contrast,
one challenge of the present analysis is to deal with the nonlinear structure of model.
21
Without this decrease, the left-hand side of (2) would reduce to the coarser bound β̂0 a(ê0 − û0 ), which
0 −û0 )∆H
may fail to provide a useful control on µ̂0 . This coarser bound guarantees that µ̂0 ≤ a(êD(û1 −û0 ) , but theright-hand side exceeds a∆H /D which may be greater than 1.

19

Defining t by t2 = 1 + a∆DH > 1 and rearranging (2) yieldsµ̂0 ≤ê0 − û0
11
= .
1 + D/a∆H û1 − û0tµ̂0 is only an average rejection probability over all choice sequences which may occur during
Block 1, but it implies (Lemma 19) the existence of a “pushdown” choice sequence for which
the posterior β̂1 at round n1 satisfiesβ̂1 ≤µ̂0 β̂0µ̂0 β̂0 + (1 − β̂0 )

≤ β̂0

t−1
= g β̂0 .
β0 t−1 + (1 − β0 )1where g = β0 +(1−β< 1, since t > 1. βn thus drops by a fixed factor along the pushdown0 )tsequence. We will construct pushdown sequences, one for each block, which succeed one
another. The choice sequence obtained from stringing the pushdown sequences together, issuch that the probability of facing H decreases geometrically across blocks.
To initiate the second block, define ê1 = uH (EH (Rn1 )) and û2 ∈ (û1 , ê1 ) by ê1 − û1 =t(û2 − û1 ), and let µ̂1 denote the probability, seen from round n1 , that H accepts only
contracts in H until û2 is reached. Let n2 denote the first round at which û2 is exceeded.

Repeating the earlier analysis, there exists a pushdown choice sequence for Block 2 such thatthe posterior β̂2 after observing this sequence until n2 satisfiesβ̂2 ≤µ̂1 β̂1µ̂1 β̂1 + (1 − β̂1 )

≤ β̂1

t−1
β̂1 t−1 + (1 − β̂1 )≤ g 2 β̂0 .

By induction, this defines a sequence of blocks k, rounds nk , values ûk , êk , and a pushdown
sequence running through these blocks such that, letting β̂k = βnk ,22β̂k ≤ g k β̂0 .

(3)As noted earlier, negotiations do not end endogenously in finite time. Instead, the backward induction argument must be constructed from a round at which the agent has not yet
been perfectly screened. This round is obtained by stopping the construction at the end
0 /(1−β0 ))of the first block, K, for which ŵK = êK − ûK < W̄ η, where W̄ = max{ t(1+b̂βt−1
(1 +

22

The actual value of uH (nk ) may be slightly above ûk , but by no more than ∆H η, by Lemma 5. Thismaximal overshoot is negligible when computing a lower bound on the number of rounds in each block,because we stop the block construction when ûk+1 − ûk is still large relative to ∆H η, as explained below.

20

+∆H
}. The constants b̂ and Ŵ are chosen to give wnK = êK − uH (nK ) and β̂K∆H ), Ŵt∆
H

appropriate lower bounds—a key ingredient for the backward induction explained below.23

The utility thresholds {ûk }k≤K guarantee a geometric decrease of wn across blocks, which

will be used to bound w0 by backward induction on k.
Lemma 7 There exists cw > 0 such thati) w0 = ê0 − û0 ≤ cwii)

wnK ≥ η.

K
t
η
t−1

(4)

To show that w0 converges to zero with η, one must bound the factor (t/(t − 1))K

appearing in (4) and, hence, find an upper bound on K. This is achieved by computing alower bound β̂ for β̂K : since β̂k is knocked down by a fixed factor g in each block, any lower
bound β̂ yields a bound on the number K of blocks needed to reach β̂.
One bound for β̂K comes from Lemma 3, which guarantees that βn ≥ β wn3 for all n, com¯
bined with Lemma 7, which implies that wnK ≥ η. This approach, described in Appendix C,
can be used to prove that w0 converges to 0 whenever the parameter t is high enough.

Proposition 4 Let t̂ denote the unique solution of 3 ln(t − 1) = 2 ln(t). If t > t̂, then forall ε > 0 there exists η̄(ε) such that w0 ≤ ε for all η ≤ η̄(ε) and corresponding PBE.

Since t2 = 1 + a∆DH , Proposition 4 applies if a is small enough and/or D is large enough.
These are the “easy” cases: Intuitively, D bounds below the loss incurred by P when he tries
to screen H. The higher this loss, the lower P’s incentive to screen H. Conversely, a bounds
the maximal gain that P can achieve from reducing H’s informational rent. The lower a,the weaker P’s incentive to screen H.
This approach has limited power, however, because it builds on Lemma 3, which only
exploits the inefficiency loss on type L to bound β̂K . Its success hinges on the relative speeds
at which βk and wk decrease across consecutive blocks, which depends on parameter values.
23

The block K must exist because ŵk converges to zero as a result of the contracts’ asymptotic efficiency(Proposition 2). The first term defining W̄ guarantees that wnK ≥ η, as shown by Lemma 7. The second

one guarantees that the number of rounds in each block k ≤ K is bounded below byûk −ûk−1 −∆H η∆H η≥W̄ η−∆H η
1> Ŵ , which can be made arbitrarily large by choosing Ŵ appropriately.
t∆H η (êk−1 −ûk−1 −∆H η) ≥
tη∆H

21

Moreover, this limitation cannot be overcome by changing the players’ utility representation,
improving the 1/3 exponent of Lemma 3, or changing the threshold t used in the construction(cf. Appendix C).
Establishing Theorem 2 for all primitives of the model requires a more sophisticatedanalysis, which exploits and combines inefficiency losses on both types to derive a sharperbound on β̂K . The key is to show the following proposition.
Proposition 5 There exist positive constants η ∗ and β ∗ such that for any PBE associatedwith η < η ∗ and any round N, βN ≤ β ∗ implies that wN < η.
Since wnK ≥ η, Proposition 5 implies that β ∗ ≤ β̂K ≤ g K β0 . This bounds K independently∗

0 /β ). Lemma 7 then implies that w0 = O(η), concluding the proof.
of η by ln(βln(−g)Part II: Proving Proposition 5
Proposition 5 provides a much sharper bound on wN than the one guaranteed by Lemma 3.
Its proof is based on a self-reinforcing phenomenon: roughly put, the smaller βn gets, andthe more persistent the inefficiency of the contract Rn becomes—whatever its current level
was—which causes βn to drop faster as more screening is needed to offset the inefficiency,
and makes the inefficiency even more persistent, etc. Since all contract sequences are asymptotically efficient, this deceleration is compatible with equilibrium only if the contract RN at
the beginning of the deceleration is sufficiently efficiency and, hence, wN is sufficiently small.
The link between βn and inefficiency persistence at the heart of this spiral will stemfrom two analytical results. First, the L-inefficiency of Rn is well captured by the index wn .
Second, while wn must converge to zero, its decrements between consecutive rounds must be√of order at most βn , owing to the geometry of the problem.24
To formalize this phenomenon, the techniques developed in this section require the useof an “orderly” choice sequence, i.e., one along which βn is decreasing and wn remains
sufficiently small across all rounds. Proving that such a sequence exists is challenging. In
this model, contracts and beliefs are not a priori well behaved: some contract choice may
24

The key step is Lemma 15, which uses analytical geometry near the efficiency curve EL and exploits

convexity of the payoff functions to derive a quadratic lower bound on the cost function as one moves awayfrom EL along L’s isoutility curves.

22create a spike up in βn or dramatically increase, at least temporarily, the inefficiency of
current contracts. Indeed, until one proves that the equilibrium starting from any contract
and belief is essentially unique (which is what Theorem 2 achieves), there may a priori be awhole range of continuation utilities for all players and types, used to sustain very complex
menu proposals and contract choices.
The formal analysis starts from the round N appearing in Proposition 5. For any roundn ≥ N and Rn+1 ∈ Mn ∪ {Rn }, let µθn (Rn+1 ) denote the probability that θ accepts Rn+1 .
Because P can immediately jump to the efficient contracts, his IC constraint implies thatβn awn ≥

X

L
βn µH
n (Rn+1 )ηD + (1 − βn )µn (Rn+1 )η(Q(Rn+1 ) − Q(EL (Rn ))) (5)Rn+1 ∈(Mn ∪{Rn })∩H=

X

µLn (Rn+1 ) [βn µn (Rn+1 )ηD + (1 − βn )η(Q(Rn+1 ) − Q(EL (Rn )))] (6)Rn+1 ∈(Mn ∪{Rn })∩H

L
25
The left-hand side of (5) is an upper bound on P’s
where µn (Rn+1 ) = µHn (Rn+1 )/µn (Rn+1 ).

gain, relative to the immediate jump, from reducing H’s rent by wn . This gain is bounded
above by a(uH (EH (Rn )) − uH (n)) = awn .26 The first term of the right-hand side captures

the inefficiency loss incurred if the agent is of type H, rejects the H-efficient contract inround n, and the breakdown occurs in round n. This loss is bounded below by D as long as
wn ≤ ε, which will be true along the choice sequence considered (cf. Lemma 4). The lastterm is the net loss if the agent is of type L and the breakdown occurs in round n.

The right-hand side of (6) is a convex combination of terms indexed by Rn+1 , so there
must exist some Rn+1 ∈ (Mn ∪ {Rn }) ∩ H such thatwn aβn ≥ βn µn (Rn+1 )ηD + (1 − βn )η(Q(Rn+1 ) − Q(EL (Rn ))).
25

(7)

We can assume without loss that µL
n (Rn+1 ) is strictly positive for all Rn+1 ∈ (Mn ∪ {Rn }) ∩ H: First,

if some contracts in this set are not chosen with positive probability, a payoff equivalent equilibrium may′
be constructed without these contracts. Second, if any contract Rn+1
is chosen only by H, Proposition 1′
implies that H gets the H-efficient contract C that gives him the same utility as Rn+1
in the next round.
′
The equilibrium can therefore be modified by having P immediately propose C instead of Rn+1
, a change

which reduces P’s cost and does not affect incentives.
26
See Lemma 13. The bound is computed using a best-case scenario for P, in which H accepts withprobability 1 the H-efficient contract Cn providing utility uH (n). It is an upper bound on P ’s gain because
Cn is the least costly way of providing H with his continuation utility.

23

Therefore, there exists a choice sequence that satisfies (7) for all n ≥ N. The analysis focuseson this sequence, which will be called an orderly sequence.

Orderly sequences are useful for the following reason. Let µn = µn (Rn+1 ) denote thelikelihood ratio associated with observing Rn+1 . Provided that wn is small enough, (7)implies that µn ≤ 1 and thus βn+1 ≤ βn . Therefore, βn decreases along any orderly sequence

as long as wn remains small enough. Even if wN is small, however, there is no a priori
guarantee that wn —and, hence βn —remain small in subsequent periods. These problems
are intertwined: from inequality (18) in Appendix A,
wn+1 ≤ wn

1 + αβn1 − βnfor some constant α. Therefore, if βn decreases sufficiently fast, wn cannot increase too much,which guarantees that βn continues to drop, and so on. This is formalized by the following
result. (All results of Part II are proved in Appendix G.)
Lemma 8 There exist η̂, β̂, and ŵ positive such that for any η < η̂ and associated PBE,
and any round N such that βN ≤ β̂ and wN ≤ Dη
, the following holds for all n ≥ N: i) βn
2a
is decreasing in n, ii) µn ≤ 3/4, and iii) wn ≤ ŵη. Moreover, we have
µn ≤

wn a
,
ηD

βn wn a ≥ (1 − βn )η(Q(Rn+1 ) − Q(EL (Rn ))).

(8)

(9)

Equation (8) shows that if P’s potential gain from screening H (proportional to wn ) is smallenough (of order η), then H must reveal himself in each round with significant probability
for the immediate jump not to be a profitable deviation. This happens when P’s maximum
screening gain over the entire continuation of the game, awn , is of the same order as the
expected loss from a breakdown in a single period. The virtue of orderly sequences is toconvert this ex ante condition into an ex post drop in P’s posterior.
Equation (9) is where the loss on L is exploited: if βn is small, it imposes an upper boundon the L-inefficiency Q(Rn+1 ) − Q(EL (Rn )) of any contract Rn+1 in the orderly sequence.

This implies that Rn+1 is very close to EL (Rn ), and it follows that H’s continuation utility

increases extremely slowly, because his “flow utility” (i.e., the utility he gets from Rn+1 if
24

a breakdown occurs in the current round) is almost indistinguishable from his continuation
utility. Formally, H’s Bellman equation implies thatuH (n + 1) − uH (n) = η(uH (n + 1) − uH (Rn+1 )),(10)

and almost-L-efficiency of Rn+1 implies that the difference on the right is arbitrarily small.
Since wn = uH (EH (Rn )) − uH (n), the difference wn+1 − wn is closely related to the left-handside of (10), which suggests that wn must decrease very slowly along the orderly sequence.
This is formalized by the next result.
Lemma 9 There exists a positive constant Aw such thatwn+1 ≥ wn − ηAwpβn+1 .

(11)

Long-run negotiations are thus dominated by two forces: First, the posterior belief βn must
decrease sufficiently fast for the screening activity to be sufficiently profitable (H-loss force).
p
Second, wn must decrease very slowly—by an order of βn+1 —because the contract must be
almost L-efficient (L-loss force). This leads to the crux of the proof: as βn decreases throughscreening, the decrements of wn become smaller in absolute value, which causes L-inefficiencyof the current contracts to persist, and forces to βn decrease faster for screening to offset the
H-inefficiency. This accelerates βn ’s drop to zero and decelerates wn ’s own drop to 0, which
may cause wn to stall altogether. Since wn must converge to zero along all sequences, this
scenario is only possible if wN was small enough to begin with.
The first part of this intuition is captured by the following lemma, which says that if wN
is too high relative to ηβN , wn ’s decreasing factor goes to 1 provided that βN lies below some2.
threshold. Let c = Da Aw and ĉ = 4Dca≥ 1.
Lemma 10 (Deceleration) If wN ≥ ĉηβN and βN ≤ c−2 /16, then lim inf n→+∞ wwn+1n
The second part of the intuition is captured by a mathematical lemma which, applied to thenratio qn = aw
, implies that wn cannot converge to 0 if its decreasing factor goes to 1.
ηD

Lemma 11 (Stalling) Consider a positive sequence {qn } and constants c′ > 0 and N ∈ N
p≥ 1. Then, {qn } doessuch that i) qn − qn+1 ≤ c′ ΠnN qk for n ≥ N, and ii) lim inf n qn+1
qn

not converge to zero.

25

Equation (11) implies that qn = awn /ηD satisfies premise i). A simple contradiction argument based on the previous lemmas then implies the following proposition. Let β̃ =min{β̂, c−2 /16}.
Proposition 6 There exists η̃ > 0 such that for any η ≤ η̃,βN ≤ β̃ and wN ≤

ηD
2a

⇒

wN ≤ ĉηβN .

, this result would prove Proposition 5 with β ∗ =If we could ignore the premise wN ≤ ηD
2amin{ĉ−1 , β̃}. Part III, below, constructs a round that satisfies this premise.

Part III: Bridging Argument.
Propositions 5 and 6 deliver a lower bound for βN at the end of block K when wN ≤ Dη2a

: if P is taking more rent away
Intuitively, a similar bound should apply if wN exceeds ηD
2afrom H, thus incurring losses on L, it better be the case that the probability of facing H is
non-negligible. To formalize this intuition, starting from wN ≥ ηD
, a natural strategy is to2a,follow a choice sequence along which wn and βn are both decreasing. When wn crosses ηD
2aProposition 6 implies that βn is greater than ĉwη n , which yields a lower bound for βn and,since the sequence {βm } was decreasing, for βN . The implementation of this idea raises twochallenges.

, it may jump to an arbitrarily low level, such as η 2 ,First, when wn drops below ηD
2arendering the lower bound ĉwη n on βn useless. To avoid this, one must select a choice sequence
along which wn ’s decrements are sufficiently small. This will be achieved by exploiting theloss on L: as noted earlier and developed below, wn+1 − wn is closely related to the L-

inefficiency of the contract Rn+1 .

Second, there need not exist a sequence along which βn and wn are both decreasing.
Instead, we will construct a sequence along which these variables cannot increase “too much,”
and break up the sequence into blocks of constant size—unlike those of Part I—across which
βn drops by a constant factor.
A similar point arose in Part II, in which we had to focus on orderly sequences, but
the approach here is different. Methodologically, it is common in the screening literature tobound a type’s average probability of rejection over all possible choice sequence realizations
26

in a block, and then extract from it one sequence which satisfies the appropriate drop in
belief. The challenge here and in Part II is that more properties are required of the sequence
extracted. In Part II, we imposed a low value on wN and used it to directly construct
an orderly sequence. Here, instead, we must first work globally, by showing that the setof sequences with the desired properties has a high enough probability in each block, anddeduce from this that one may extract a sequence within that set that also satisfies thestandard drop in belief.
To analyze this problem, it suffices to consider the case wN ≤ W̄ η, which was used todetermine the last block of Part I.27 To address the first challenge, we introduce the variable
yn = uH (EH (Rn )) − uH (Rn+1 ), represented on Figure 4. yn provides the control required on
the decrements of wn : subtracting uH (EH (Rn )) from H’s Bellman equation (equation (10)in Part II) and rearranging it yields
wn+1 = wn − ηyn + ηwn+1 + (1 − η)(uH (EH (Rn+1 )) − uH (EH (Rn ))).

(12)

The control of wn ’s decrements through yn stems from the following lemma (all proofs forthis part are in Appendix F). Fix a positive integer N̄ and a small positive constant ε̄.
Lemma 12 There exist positive constants ky , kw , and kε with the following properties. Consider any round n̄ such that wn̄ ≤ W̄ η and βn̄ ≤ ε̄N̄ , and let S denote the event that theagent chooses in all rounds n ∈ {n̄, . . . , n̄ + N̄ − 1} contracts such that yn ≤ ky ε̄1/4 , βn ≤ ε̄,√and wn ≤ kw η. For η small enough, the probability of S is greater than (1 − kε ε̄)2N̄ .
Lemma 12 implies, for ε̄ and βn̄ small enough, that choice sequences along which yn is lessthan any fixed amount for at least N̄ rounds have probability almost 1. We now constructblocks of fixed size N̄, to be set shortly. The first block starts at round N, the second one at
N + N̄ , etc. In the first round, n̄, of any given block, P’s IC constraint implies that, lettingen̄ = uH (EH (Rn̄ )),27

βn̄ a (1 − µn̄ )(en̄ − uH (n̄)) + µn̄ (en̄ − E[uH (n̄ + N̄ )]) ≥ βn̄ µn̄ Dη N̄

If wN lies above this value, the block construction of Part I can be used to decrease βn block by block

until reaching W̄ η. Any lower bound on the posterior β̂K once W̄ η is reached also applies to βN at the
beginning of the block construction, since β decreases across blocks.

27

x2
EH

EL

uH (EH (Rn))E (R )HnwnuH (n)EL (Rn)ynuH (Rn+1)

Rn+1

Rn

x1

Figure 4: Representation of yn and wn .
where µn̄ is the probability, seen from round n̄, that H rejects all H-efficient contracts
between rounds n̄ and n̄ + N̄ . The argument behind this inequality is similar to the oneused in Part I, the main difference being the expectation of uH (n̄ + N̄) at the end of theblock, which is no longer controlled. The loss bound D is valid by virtue of Lemma 4 andthe fact that wn̄ is of order η and hence less than ε, for η small enough, and the probabilityof a breakdown during the block is 1 − (1 − η)N̄ ∼ η N̄. This implies that
µn̄ ≤

aW̄
a(en̄ − uH (n̄))≤,
a(E[uH (n̄ + N̄)] − uH (n̄)) + Dη N̄
D N̄

(13)

where the second inequality stems from the inequality EuH (n̄+ N̄ ) −uH (n̄) ≥ 0, which holds
by monotonicity of uH (n), and the fact that wn̄ = en̄ − uH (n̄) ≤ W̄ η.28

These blocks are simpler than those of Part I because their size N̄ is fixed and H’s utility

at the end of each block is “free.” Two reasons motivate this difference: First, H’s utility
increments are now so small that they no longer provide a useful lower bound on the number
28

This inequality holds without loss of generality as explained in Remark 3 of the Online Appendix.

28

of rounds in each block. Second, P’s potential gain from screening H is now so small that it
is no longer useful to distinguish how much rent can taken from H across different blocks.29
N̄ is set so that aW̄ /D N̄ and, hence, µn̄ , be less than 41 . As in Part I, µn̄ is an average
probability over possible choice sequences. And as in Part I, we wish to select a pushdown
sequence along which the posterior at the end of the block is pushed down by this probability.
However, here we also wish to select the sequence so that yn remains small. Fortunately,
Lemma 12 guarantees that these sequences have probability almost 1 provided that ε̄ is smallenough, and hence guarantees the dual goal of having small decrements for wn and reducing
the posterior βn across blocks. Formally, starting from N, let M denote the first round atwhich wn drops below ηD/2a. The next result guarantees that wM is bounded below andthat βM is bounded above in fixed proportion to βN .
, W̄ η) and βN ≤ β̌,
Proposition 7 (Bridge) There exist kβ and β̌ such that if wN ∈ ( ηD
2athen there is a choice sequence such that 1. wM ≥ ηD
and 2. βM ≤ kβ βN , where M ≥ N is
4a.
the first round such that wM ≤ ηD
2a

Proposition 5 follows easily: provided that βN lies below some appropriate threshold related
to kβ and β̌, βM will be small enough to apply Proposition 6 of Part II to round M. Since,
also, wM is bounded below by Part 1 of Proposition 7, this provides an η-independent lower
bound for βM and, hence, for βN . The proof is in Appendix H.

6Discussion

6.1

Related Literature

Contract renegotiation with private information has traditionally been studied from two
perspectives. The first one is axiomatic and focuses on “renegotiation-proof” contracts.30 It
essentially assumes that renegotiation leads to an efficient contract, ignoring the difficulties
29

Also note that the bound computed in (13) uses the fact that wn̄ ≤ W̄ η, but if wn̄ only satisfied an

η-independent bound, as in Part I, the right-hand side of (13) would be of order 1/η and thus useless.
30
See Dewatripont (1989), Maskin and Tirole (1992), Battaglini (2007), Maestri (2015), and Strulovici(2011, 2013). A similar approach has been used to study renegotiation in repeated games with complete
information. See, e.g., Bernheim and Ray (1989) and Farrell and Maskin (1989).

29

which may arise when some party holds private information. The second one focuses on
simple renegotiation protocols, in which the principal makes a single take-it-or-leave-it offer.
This perspective typically results in inefficient contracts.31 The second perspective seems
incomplete: what, in reality, prevents the principal from proposing a new contract after
learning the inefficiency of the current contract? Such a restriction amounts to a strong
form of commitment for the principal and can even result in full-commitment outcomes.32
This paper shows that, by dropping the restriction on the number of negotiation rounds, one
can reconcile these two perspectives.33
While the model’s main interpretation concerns the implementation of a single contract
arising from a sequence of temporary agreements, it has an alternative interpretation: In
each round, the current contract is implemented and parties receive the corresponding payoff,and the interruption probability is reinterpreted as the discount rate between periods.34 In
each period, the principal has an opportunity to renegotiate the contract for future periods.
The model is formally equivalent to an infinite-horizon version of Hart and Tirole (1988),
with divisible goods as in Laffont and Tirole (1990), arbitrary utility and cost functions, andin which the contract is constant until renegotiated. Thus interpreted, this paper shows that
all equilibria become efficient as the discount rate goes to zero.
Maestri (2015) analyzes renegotiation-proof equilibria in a divisible-good, binary-typeversion of the Hart-Tirole framework and proposes a renegotiation-proofness refinement of
PBE, formulated recursively: In each period, the contracts offered by the principal maximize his revenue among all renegotiation-proof continuations, given the current contract and
31
32

See Hart and Tirole (1988) and Fudenberg and Tirole (1990).
For example, imposing any finite number k of negotiation opportunities results in the full-commitment

outcome: the principal simply passes the first k−1 opportunities to negotiate the contract, and then proposes
the full commitment allocation in the last round. Wang (1998) considers a more flexible protocol, in which
the principal proposes contracts until an agreement is reached. Such protocol leaves a high commitment
power to the principal, since he cannot renegotiate any agreement. Indeed, Wang shows that the principal
achieves the full commitment allocation, which is also ex post inefficient. See also Beaudry and Poitevin(1993) and Matthews (1995).
33
Brennan and Watson (2013) study another friction in the form of explicit renegotiation costs. Anotherextension of the model would consider general negotiation protocols, as in Rubinstein and Wolinsky (1992).
34
The formal equivalence appears clearly in the payoff formulas of page 8: 1 − η plays the role of the
discount factor, and η is the weight put on the current period.

30

beliefs. The principal can propose at most two contracts in each period and the probabilityof facing the high type decreases monotonically as long as this type is not fully revealed.
As the discount rate goes to zero, all renegotiation-proof equilibria become efficient. By
contrast, this paper implies that all PBEs become efficient, allowing for arbitrarily large
menus of contracts, general utility and cost functions, and an infinite horizon, and focusing
on contracts which are constant until renegotiated.35
Finally, the paper is related to the literature on bargaining and reputation, in which someplayers are trying to determine whether other players have a “commitment” type.36 This
paper presents specific challenges described in Section 5 and differs from reputation models
in other ways: i) the “actions” of the players are endogenous, because the principal sets thecontract menu in each round, ii) the state space includes the current contract, in addition
to the principal’s belief, and iii) all types are strategic.37

6.2

Extensions

The analysis has focused on two types. With more types, it is natural to conjecture that ex
post efficiency still obtains. Appendix L explains how the techniques developed here can be
35

Conceptually, the papers’ objectives are complementary: while Maestri studies renegotiation-proof equi-

libria when trade happens over time, the present paper shows—under its main interpretation—that if parties
have frequent opportunities to bargain before a one-time trade decision, all equilibria will be approximately
efficient. It thus provides a non-cooperative foundation for renegotiation-proof contracts. The technicalcontributions are also different. For example, an important part of Maestri’s analysis lies in its conceptual
work to define renegotiation-proof contracts. By contrast, this paper analyzes all equilibria, which creates
important challenges described in earlier sections.
36
Fudenberg and Levine (1989), Schmidt (1993), Abreu and Gul (2000), Cripps et al. (2005), and Atakan
and Ekmekci (2012, 2014). Abreu and Pearce (2007) consider a repeated game setting in which players
bargain over commitments for the continuation of the game.
37
The richer state space and the nonlinear geometry of the problem (the parties’ utility functions are
nonlinear) make the problem particularly difficult to analyze. Some differences can formally be incorporated
into the standard reputation framework. For example, the action space of the agent may be assumed tobe fixed by setting a large negative default value if the agent chooses anything outside of the principal’s
proposed set. This formal equivalence does not resolve the substantive differences between the settings.

31

used to make progress in this direction.38
The model featured, under its main interpretation, a single “delivery” time at which
the contract is implemented. To provide a foundation for renegotiation-proof contracts
with multiple deliveries, one should consider a more general model with multiple “physical”
events (payments or efforts are made, exogenous information arrives, etc) and renegotiation
protocols like the one studied here inserted between consecutive physical events. Each renegotiation protocol would then pertain to continuation contracts over the remaining horizon,
and any event arrival would trigger the end of current negotiations.

AppendicesA

Inequalities

This appendix presents key inequalities used in other appendices and derived in Appendix E.
Lemma 13 (Regularity Bounds) There exist positive constants a, a, b, b such that for¯ ¯
any C, Ĉ ∈ EH such that uH (C) < uH (Ĉ),
a(uH (Ĉ) − uH (C)) ≤ Q(Ĉ) − Q(C) ≤ a(uH (Ĉ) − uH (C))¯
b(Q(Ê) − Q(E)) ≤ Q(Ĉ) − Q(C) ≤ b(Q(Ê) − Q(E)),
¯

(14)
(15)

where E (resp. Ê) is the L-efficient contract that gives H the same utility as C (resp. Ĉ).
Let QL denote P’s expected cost, view from round n, conditional on facing L.
Lemma 14 (Incentive Bounds) Given any PBE and choice sequence {Rn }, there exist

positive constants α, γ, b, and b̂ such that, for any n,

βn
awn ,
(1 − βn )αβnwn ,uH (EH (Rn+1 )) − uH (EH (Rn )) ≤1 − βn
αβnwn+1 ≤ wn 1 +,1 − βnuL (Rn ) − uL (Rn+1 ) ≤ γβn+1 wn+1 ,QL ≤ Q(EL (Rn )) +wn+1 (1 + bβn+1 ) ≥ wn − ηyn ,
38

(16)
(17)
(18)
(19)
(20)

With three types H, M , and L, for instance, the contracts {Eθ (R0 )}θ∈{H,M,L} would be the least costlyθ-efficient contracts which are incentive compatible and individual rational, starting at R0 .

32

and, for any n < n′ ,uH (EH (Rn′ )) − uH (EH (Rn )) ≥ −b̂βn′w n′ .
1 − βn′(21)

Lemma 15 (Geometric Bound) There exists q > 0 such that for any C ∈ EL and R ∈ H¯
such that uL (R) = uL (C),
Q(R) − Q(C) ≥ q (uH (C) − uH (R))2 .
¯Lemma 16 There exist positive constants k2 and k3 such thatyn2 ≤ k2 [Q(Rn+1 ) − Q(EL (Rn ))] + k3 (max{(βn wn /(1 − βn ))2 , (βn+1 wn+1 )2 } + βn+1 wn+1 ) (22)B

Proofs of Section 4

Proof of Proposition 1
Part i) Let ū denote the agent’s supremum over his expected utility, given his type θ,over all possible continuation PBEs starting from R0 at which P puts probability 1 on type
θ, and let u = uθ (R0 ). Suppose by contradiction that ū > u. By time homogeneity, ū will be
the same in the next round if the agent rejects new offers from P in round 0 and P continues
to assign probability 1 on facing type θ. In such case, the agent’s continuation payoff isbounded above by ũ = ηu + (1 − η)ū < ū. Consider any PBE that gives θ an expected utility
u0 ∈ (ũ, ū) (such PBE must exist, by definition of ū). Suppose that the principal deviates

by proposing the θ-efficient contract C that give θ a utility level u′ in (ũ, u0 ). By definitionof a PBE,39 P continues to assign probability 1 to type θ after his own deviation. If theagent accepts C with probability 1, the deviation is strictly profitable to P since C is theleast costly way of providing utility u′ < u0 to the agent. If the agent rejects the offer withpositive probability, then by Bayes rule, P must continue to assign probability 1 to type θ,which implies that his continuation utility is bounded above by ū. Therefore, the agent’s
rejection is strictly suboptimal, implying that the agent must accept C with probability 1
and the deviation is profitable.40 Let Q denote the cost of the θ-efficient contract, C , that
¯
¯
39
40

See Fudenberg and Tirole (1991), part iii) of the definition.
The continuation play after P’s deviation must be a PBE of the corresponding continuation game.

Therefore, if θ’s continuation strategy, after P’s deviation, is to reject the proposed deviation with positive
probability, Bayes rule applies. I am grateful to Marcin Peski for proposing the current version of thisargument.

33

provides utility u to θ. Clearly, any PBE must cost exactly Q, otherwise P has a profitable¯
deviation which is to propose the θ-efficient contract that gives θ slightly more than u andcosts less than following the PBE. Moreover, the only way of achieving Q is to propose C in¯¯
the first round and have it accepted with probability one.
Part ii) Suppose without loss that θ = L (the opposite case is treated identically). Let
uL = uL (R0 ) and uH = uH (R0 ). Also let ūH (β) denote the supremum utility that H canachieve over any continuation PBE starting from R0 when P assigns probability β to H,
and let ūH = supβ∈[0,1] ūH (β). Suppose by contradiction that ūH > uH . Then, for any
small ε > 0, there exists β̄ and an associated PBE for which H’s continuation utility is
above ūH − ε > uH . For that PBE, because L gets at least uL and R0 is L-efficient, wehave Q̄L ≥ Q where Q = Q(R0 ) and Q̄L is P’s expected cost in that PBE conditional on

facing θL . Since not proposing any new contract is always feasible for P, and costs Q, thecontinuation cost Q̄H conditional on facing H must satisfy Q̄H ≤ Q to offset the weakly

higher cost conditional on facing L. Suppose that P deviates from that PBE by proposing
the H-efficient contract that gives θH utility ūH − ε − ǫ, for arbitrarily small ǫ. Because,

for small enough ε and ǫ, ūH − ε − ǫ > ηuH + (1 − η)ūH , H accepts this proposal withprobability 1. For any strategy that θL chooses and continuation equilibrium, this proposal
strictly reduces P’s expected cost (since Q̄H ≤ Q), yielding a contradiction. This shows thatūH (β) = uH for all β.41 To conclude, suppose that P proposes the H-efficient contract thatgives H utility uH + ǫ̃, for ǫ̃ arbitrarily small. From the previous observation, H must accept
that contract regardless of L’s strategy. This shows that P can and, hence, does achieve thefull-commitment optimal cost under any PBE. This proves Part ii).
Part iii) Suppose that QL ≥ QH where Qθ = Q(Eθ (R0 )) (the opposite case is proved

symmetrically) and let Q̄ denote the maximal expected cost incurred by P over all PBEs
and beliefs β ∈ [0, 1], starting from R0 .

We start by showing that Q̄ ≤ QL . Suppose by contradiction that Q̄ > QL and consider

any PBE that achieves Q̄.42 Now suppose that P deviates by proposing the pair C̃L , C̃H of41

If β = 0, P does not propose anything new, from i) and L-efficiency of R0 , and the result trivially holds

in that case too.
42
If the supremum Q̄ is not achieved, the argument below can easily be adapted by considering a PBE
whose expected cost is arbitrarily close to Q̄.

34

contracts such that C̃θ is efficient for θ and costs Q̄ − ε for some ε arbitrarily small comparedto η. Those contracts maximize each type’s utility subject to costing P at most Q̄ − ε.

Because these contracts are efficient and incentive compatible, Part ii) guarantees that notype ever chooses the contract meant for the other type. Moreover, no matter what belief
and continuation PBE follows rejection of these contracts, P’s continuation cost must be lessthan Q̄, by definition of Q̄. But this latter bound implies that there must be at least one typeθ of the agent who is getting a lower payoff if he rejects C̃θ than if he accepts it: conditionalon rejection P has to be spending weakly less on at least one type of the agent than underC̃θ (up to ε, which is negligible compared to η). Moreover, the contract C̃θ maximizes thistype’s utility subject to P spending less than C̃θ . Since rejection leads to a renegotiation
breakdown with probability η, giving this type a strictly lower utility than C̃θ , it is strictly
better for this type to accept C̃θ with probability 1. As a result, a rejection fully reveals thatthe agent is of the other type. From Part i), that type agent gets uθ (R0 ) after the rejection,
which is strictly less than the utility he gets from C̃θ —since this contract maximizes theagent’s utility subject to a higher cost than what P incurs with R0 . Therefore, both types
accept their contract, and this reduces the cost of the principal strictly below Q̄, showingthat this is a profitable deviation. Thus, necessarily, Q̄ ≤ QL .

Since L gets utility at least uL (R0 ) in any PBE, and QL is the least cost for providing

this utility, in all PBEs starting with β ∈ (0, 1), P must spend weakly less than Q̄ on H

to guarantee that Q̄ ≤ QL . Let ūH denote the supremum expected utility that H getsover all PBEs and beliefs β > 0. Since P spends less than QL on H, ūH is bounded by

the utility ûH obtained from the H-efficient contract ĈH that costs QL . We will show thatūH = uH (EH (R0 )). Suppose by contradiction that ūH > uH (EH (R0 )), and consider a PBE
that achieves ūH .43 The expected cost Q from that PBE must be above βQ(C̄H ) + (1 − β)QL

where C̄H is the H-efficient contract that gives utility ūH to H. Suppose that P deviates by
proposing the contracts C̃L , C̃H such that C̃L is L-efficient and gives utility uL (C) + ε2 to L
and C̃H is H-efficient and gives utility ūH −ε to H, for ε small compared to η. H accepts C̃H ,since rejection leads to a continuation utility bounded above by ūH and to a strictly lower

payoff in case of a breakdown. L then also accepts the contract since rejecting it would reveal
43

Again, the proof is easily adapted if the supremum is not achieved, by considering a PBE that gets very

close to providing ūH .

35his type and, by Part i), yield a utility of uL (EL (R0 )). The cost reduction facing H is oforder ε relative to Q(C̄H ), while the cost increase facing L is of order ε2 relative to QL . Thisdeviation is thus strictly profitable for ε small enough, which shows that ūH = uH (EH (R0 )).
Proceeding as in the end of the proof of Part i), this shows that L’s maximal utility acrossall PBEs for β ∈ (0, 1) is uL (EL (R0 )).

Part iv) The proof is similar to that of Part iii). Let Q̄ denote P’s maximal expected

cost over all PBEs and beliefs, starting from R0 . We will start by showing that Q̄ ≤ Q(EL ),where EL = EL (R0 ). Suppose by contradiction that Q̄ is strictly greater than Q(EL ) andachieved for some PBE and belief,44 and consider the following deviation: P proposes thecontracts C̃θ that are efficient for each type and cost Q̄ − ε for ε arbitrarily small. It iseasily shown that these contracts are IC, and by a similar argument to the one used in Partiii), rejecting these contracts is a strictly dominated strategy for one of the two types, andhence for both types. This is a strictly profitable deviation for P, yielding a contradiction.
Hence, Q̄ ≤ Q(EL ). Since L’s expected utility is at least uL (R0 ) in all PBEs, and providing

this utility costs at least QL = Q(EL ) to P, this means that P spends at most QL on H,
in all PBEs, and for all initial beliefs β > 0. This implies that H’s expected utility isbounded above by the utility that he achieves with the H-efficient contract that costs QL .
We now show that H’s expected utility is bounded above by uH (EL ). Suppose not andconsider a PBE that gives H his highest utility across all PBEs and beliefs, which we denoteūH > uH (EL ). The expected cost Q from this PBE must exceed βQ(C̄H ) + (1 − β)QL ,where C̄H is the H-efficient contract that gives utility ūH to H. Suppose that P deviates by

proposing the contracts C̃L , C̃H such that C̃L is L-efficient and gives utility uL (R0 ) + ε2 toL, and C̃H is H-efficient and gives utility ūH − ε to H, for ε arbitrarily small. Because C̃H

gives strictly more to H than ūH , H accepts C̃H and, hence, L accepts C̃L . As in Part iii),
this deviation is strictly profitable and yields the desired contradiction. Also note that for εsmall enough, ūH > uH (EL ), so mimicking L is suboptimal for H.45
Proof of Lemma 1
The result follows from Part iv) of Proposition 1: EH (Rn ) plus any small amount gives
44
45

As before, one can use a PBE that yields a cost arbitrarily close to Q̄, in case it is not exactly achieved.
It is straightforward to show that L does not want to mimic H, since P spends less on H than on L,

and L is already getting his maximal utility given the cost that P incurs conditional on facing L.

36

a strictly higher utility to H than his maximal continuation utility and strictly more utility
than EL (Rn ). Therefore, H accepts the contract with probability 1. Since L strictly prefers
EL (Rn ) to EH (Rn ) (Part ii) of Proposition 1) and his type is revealed unless he takes thestrictly suboptimal contract EH (Rn ), L accepts EL (Rn ).



Proof of Lemma 2
Consider any PBE starting with R0 ∈ H and, by contradiction, the first round n suchthat i) Rn is in H and ii) L accepts with positive probability a contract Rn+1 that is ina different configuration. Suppose first that Rn+1 is in the No-Rent configuration. ThenuL (n) = uL (Rn+1 ), by Part iii) of Proposition 1, which implies that uL (Rn ) ≤ uL (Rn+1 ):Rn+1 is on a weakly higher isoutility curve of uL than Rn . Moreover, because H can always

accept Rn+1 , uH (n) ≥ uH (Rn+1 ) > uH (EH (Rn )), where the strict inequality comes fromthe fact that uH is increasing along the isoutility curve of uL in the direction of EH .46 Thisimplies that the continuation cost for P is strictly above βn Q(EH (Rn )) + (1 − βn)Q(EL (Rn )),which contradicts Lemma 1. Now suppose that Rn+1 is in the L-Rent configuration. Partiv) of Proposition 1 applied to the L-Rent configuration implies that, by choosing Rn+1 ,L gets a continuation utility of at most uL (ẼL (Rn+1 )) where ẼL (R̃) is defined—when R̃ isin the L-Rent configuration—similarly to EH (R) when R is in the H-Rent configuration.
Therefore, uL (ẼL (Rn+1 )) must be weakly greater than uL (Rn ). However, notice that whenẼL is constructed, we use L’s isoutility curve between the efficiency curves EH and EL , whichis steeper than H’s isoutility curve at ẼL (Rn+1 ), from the single-crossing property. As canbe easily checked graphically, this implies that uH (Rn+1 ) must have been strictly greaterthan uH (EH (Rn )), contradicting Part iv) of Proposition 1 applied to H.



Proof of Proposition 2i) Observe, first, that negotiation cannot end endogenously at a finite round N with
βn = βN > 0 and Rn = RN ∈ H for all n ≥ N. If this were the case, P could strictly reduce
his cost at round N by proposing the H-efficient contract EH (RN ) and have it accepted by H

with probability 1, by Part iv) of Proposition 1. Suppose instead that P keeps proposing newcontracts until renegotiation is exogenously interrupted, and suppose by contradiction that
there is a choice sequence with an associated belief subsequence {βn(k) }k∈N that converges
46

More explicitly, we have uH (Rn+1 ) > uH (EL (Rn+1 )) ≥ uH (EL (Rn )) = uH (EH (Rn )).

37

to β ∗ > 0 (so both types accept each contract in this subsequence with strictly positive
probability). Let u∗H = sup{uH (Rn )} where the supremum is taken over all contracts in the
choice sequence. For H to accept Rn with positive probability infinitely often, uH (Rn ) must
converge to u∗H for any subsequence, including along the subsequence {n(k)}.47 However,this implies that proposing the H-efficient contract CH that gives u∗H to H is a strictly
profitable deviation as βn(k) gets arbitrarily close to β ∗ : it does not change P’s cost conditional
on facing L but it strictly reduces P’s expected cost by an amount arbitrarily close toβ ∗ [Q(CL ) − Q(CH )], where Cθ is the θ-efficient contract that provides H with utility u∗H .48

ii) Suppose that there exists ε > 0 and a subsequence of rounds, indexed by m, for which

Q(Rm ) − Q(EL (Rm )) ≥ ε. For m large enough, βm converges to zero, from part i), and isηεthus bounded above by 2∆, where ∆Q = maxC∈C Q(C) − minC∈C Q(C). Therefore, P canQdeviate by proposing EL (Rm ), EH (Rm ), which are respectively accepted by L and H. Thison H, given the upperdeviation yields an immediate gain of ηε on L and a loss of at most ηε2bound on βm , and is thus strictly profitable. This shows that the limit points of {Rn } are all
L-efficient. Let u∗L = sup{uL (Rn )}. There is a subsequence indexed by m̃ for which uL (Rm̃ )converges to u∗L . Moreover, since L can always hold on to any contract Rn along the choice
sequence, and thus in particular to the contracts occurring along the subsequence {Rm̃ },uL (Rn ) must converge to u∗L for all subsequences. Combining these observations, {Rn } must
converge to the L-efficient contract C̄L such that uL (C̄L ) = u∗L .

iii) Parts i) and ii) have shown that βn converges to zero and Rn converges to an Lefficient contract as n goes to infinity. This implies that EH (Rn ) gives asymptotically thesame utility to H as Rn does and, hence, that wn converges to zero.49
Proof of Lemma 3
47u∗H

Otherwise, there must exist a subsequence of rounds for which uH (Rm+1 ) is bounded above away fromby some constant δ > 0. However, H’s continuation utility, uH (m), is nondecreasing and becomes

arbitrarily close to u∗H . (Monotonicity comes Lemma 5.) When H’s continuation gets within εη of u∗H forsome ε arbitrarily small, this implies that accepting Rm+1 causes a loss of order ηδ, due to the probabilityof an immediate breakdown, and contradicts the fact that uH (m) is within εη of u∗H .
48
Since uH (Rn ) gets arbitrarily close to u∗H and Rn lies in H, Q(Rn ) becomes arbitrarily close to (orabove) Q(CL ) as n gets large.
49
Put differently, the contract C̄H defined in Part iii) of the proof of Proposition 1 satisfies uH (C̄H ) =uH (C̄L ).

38

Without loss, set n = 0, β = β0 , R = R0 , ūL = uL (R0 ), ūH = uH (R0 ), and Q̄ =Q(EL (R0 )). It suffices to prove the claim for β ≤ β́ for some small threshold β́ ∈ (0, 1).50

Focusing on this case, (16) implies that P’s expected cost QL conditional on facing L satisfies
H. For any small ǫ > 0, to be chosen shortly, let T ǫ = {R ∈ H :QL ≤ Q̄ + q́β where q́ = a∆1−β́uH (EL (R)) − uH (R) ≤ ǫ} and D ǫ = H \ T ǫ . Graphically, T ǫ represents a tube-like region of
H bordering EL , whose (varying) width is of order ǫ.

Let τ denote the index of the round immediately following the exogenous breakdown—thecontract implemented is thus Rτ , recalling that the agent chooses contract Rn+1 in periodTn. Let pL denote the probability that Rτ ∈ D ǫ , conditional on facing type L, and uD
L and uLdenote L’s expected utilities conditional on Rτ ∈ D ǫ and Rτ ∈ T ǫ , respectively. Similarly,Tǫǫ
let QD
L and QL denote P’s expected cost conditional on facing L when Rτ ∈ D and Rτ ∈ T ,T
Drespectively. We have (1−pL )uTL +pL uD
L = uL (0) ≥ ūL and (1−pL )QL +pL QL = QL ≤ Q̄+q́β.

By definition of D ǫ , Lemma 15 implies,51 whenever Rτ ∈ D ǫ , that
Q(Rτ ) ≥ Q(EL (Rτ )) + q ǫ2 .
¯

(23)Taking expectations in (23) conditional on Rτ ∈ D ǫ yields
ǫ2
QD
L ≥ E[Q(EL (Rτ ))|Rτ ∈ D ] + q ǫ .
¯Let EL (u) denote the L-efficient contract that provides L with utility u. By convexity ofQ(·) and concavity of uL (·),
E[Q(EL (Rτ ))|Rτ ∈ D ǫ ] ≥ Q(EL (uD
L ))since each EL (Rτ ) gives L the same utility as Rτ and, conditional on lying in D ǫ , the contractsRτ ’s give L a utility of uD
L in expectation. Again by convexity of Q(·) and concavity of uL (·),
T
we have Q̄ ≤ pL Q(EL (uDL )) + (1 − pL )Q(EL (uL )) since Q̄ is the smallest cost for providing50

wn is uniformly bounded as a difference of bounded utilities and we can always increase Kw so thatKw β́ 1/3 exceeds wn ’s upper bound.
51
Here, as well as later in the proof, we are applying an inequality to round τ . The inequality must holdbecause the variable Rτ is determined before the breakdown, by the agent’s contract choice at round τ − 1

before knowing whether the breakdown will occur at the end of that round. More generally, any inequality
satisfied by variables determined before the breakdown must hold regardless of whether the breakdown occurs
immediately after this determination or later.

39

L with utility ūL , while the right-hand side is the cost associated with a particular way ofTproviding L with utility uL (0) = pL uD
L + (1 − pL )uL ≥ ūL . Combining these inequalities,
2T2Q̄ + q́β ≥ QL ≥ pL (Q(EL (uDL )) + q ǫ ) + (1 − pL )Q(EL (uL )) ≥ Q̄ + pL q ǫ ,
¯
¯

(24)

which implies that pL qǫ2 ≤ q́β. Choosing ǫ = β 1/3 , we get pL ≤ kp β 1/3 where kp = q́/q > 0.
¯¯
The reason for choosing this value of ǫ comes from (27) below: it optimizes the trade-off
between a higher probability of Rτ being in T ǫ and a tighter bound on w over T ǫ .
Set T = T β

1/3

and D = D β

1/3

and let p denote the unconditional probability that Rτ ∈/T.

We have p = βpH + (1 − β)pL , where pH is the probability that Rτ ∈/ T conditional on theagent being of type H,52 and thusp ≤ β + pL ≤ kp β 1/3(25)where the last inequality is obtained by increasing the constant kp derived earlier, whose
precise value is unimportant for the proof.
The breakdown time τ has finite expectation: E[τ ] = 1/η. By the optional sampling theorem, this implies that E[βτ ] = β. Moreover, by definition of p we have E[βτ ] = pE[βτ |Rτ ∈/T ] + (1 − p)E[βτ |Rτ ∈ T ]. Since p ≤ kp β 1/3 , this implies thatE[βτ |Rτ ∈ T ] ≤ β + O(β 4/3 ).

(26)

Suppose βτ ≤ β̄ for some β̄ ∈ (0, 1). Applying (21) to rounds n = 0 and n′ = τ ,uH (EH (Rτ )) − ūH ≥ −

b̂β̄
wτ .
1 − β̄

When Rτ ∈ T , we havewτ = uH (EH (Rτ )) − uH (τ ) ≤ uH (EL (Rτ )) − uH (Rτ ) ≤ ǫ = β 1/3 ,(27)

where the first inequality comes from uH (EL (Rτ )) = uH (EH (Rτ )) and uH (τ ) ≥ uH (Rτ ).53

and the second one follows from the definition of T .
52

If the agent is of type H, Rτ may also be an H-efficient contract, hence the conditioning Rτ ∈/ T rather

than Rτ ∈ D, as these events are only equivalent for type L.
53
Recall that uH (Rn ) ≤ uH (n) for all n, since holding to Rn forever after round n is always a feasible
strategy for H.

40

Moreover, for Rτ ∈ T we have uH (Rτ ) − uH (EH (Rτ )) ≥ −ǫ = −β 1/3 . Letting k̄ =1 + b̂ 1−β̄ β̄ > 0, these observations imply that for βτ ≤ β̄,uH (Rτ ) ≥ ūH − k̄β 1/3 .

(28)

Let p̄ denote the probability, conditional on Rτ ∈ T , that βτ ≥ β̄. We havep̄β̄ ≤ p̄E[βτ |βτ ≥ β̄, Rτ ∈ T ] + (1 − p̄)E[βτ |βτ < β̄, Rτ ∈ T ] = E[βτ |Rτ ∈ T ] ≤ β + O(β 4/3 ).
Therefore, p̄ ≤ k̄β for some constant k̄ > 0. Combining this with (25), the unconditional
probability that Rτ ends up in T and that βτ ≤ β̄ is bounded below by(1 − kp β 1/3 )(1 − k̄β) = 1 − kp β 1/3 + O(β).

(29)

Thus, either H or L has probability at least 1 − kp β 1/3 of ending up with a contract

in T and a breakdown belief βτ ≤ β̄. If it is H, (28) implies that H has a probability atleast 1 − kp β 1/3 of getting a utility uH (Rτ ) ≥ ūH − k̄β 1/3 . If it is L, H can mimic L forthe entire game and guarantee himself the same utility with probability at least 1 − kp β 1/3 .

Either way, H’s expected utility at the beginning of the game satisfies uH (0) = E[uH (Rτ )] ≥pm (ūH − k̄β 1/3 ) + (1 − pm ) minC∈C uH (C) where pm ≥ 1 − kp β 1/3 . The right-hand side isbounded below by ūH −Kw β 1/3 for some Kw > 0. Since w0 = ūH −uH (0), the lemma follows.

C

Proof of Proposition 4

Lemmas 3 and Lemma 7 imply that β̂K ≥ β η 3 for some β > 0. The condition β̂K ≤ g K β0¯¯t
, Lemma 7 implies thatthen implies that g K ≥ β η 3 /β0 . Defining ρ by g −ρ = t−1¯
w0 ≤ cw g −ρK ≤ c′w η 1−3ρ .
This shows that w0 converges to 0 with η if ρ < 1/3. By definition of g and ρ, we have
ρ = ln(t/(t − 1))/ ln(β0 + t(1 − β0 )). For β0 small, ρ is thus close toρ0 = 1 −ln(t − 1)
ln t

The RHS is decreases in t ∈ (1, +∞) from +∞ to 0. Therefore, there is a threshold t̂ above

which the condition ρ0 < 1/3 is satisfied, provided that β0 is less than some level β0 (t) > 0.
41

Using the block structure of Part I, starting from any β0 < 1, we reach β0 (t) in at mostK0 blocks, where K0 is defined by g0K0 = β0 /β0 (t) and g0 = 1/(β0 + t(1 − β0 ). w0 is thustbounded above by t−1

K0 1−3ρη, which converges to 0 as η goes to zero, proving the result.

The 1/3 exponent coming from Lemma 3 was chosen optimally, as explained in thelemma’s proof. Moreover, any bound of the form wn ≤ βnα for some α > 0 would merely

change ρ’s threshold; it would not cover all primitives of the model.

It may be possible to improve the value of t chosen in Part I, so as to obtain a lower valueof ρ0 . However, the minimal value of ρ0 over all possible values of t diverges to +∞ as 1+ a∆DH
goes to 1, as shown in Appendix F.1. Low values of 1 + a∆DH create two problems: the factor
g controlling the belief decrease goes to 1, becoming useless, and the factor t/(t − 1) used tobound w0 above in Lemma 7 becomes arbitrarily large. Using the approach Proposition 4seems hopeless when screening incentives are high for all values of t.
Finally, players are expected-utility maximizers whose utility functions may, in some applications, be defined up to an affine transformation. Could one get more from Proposition 4
by changing players’ utility representation? The answer is negative: scaling or translating
the utility and cost functions has no impact on the key parameter t.54

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